Is what I did, completing the recitation of Al-Fatiha and then following the Imam in the Funeral Prayer, correct, and what should I do if I encounter this situation again in the future?

Reciting Al-Fatiha is a pillar of the funeral prayer according to some scholars. If the Imam says the second Takbir before you finish Al-Fatiha, then stop reciting it and follow him in the Takbir; for he bears the rest of Al-Fatiha on your behalf.